If $|a| = 4$,$|b| = 2$ and the angle between $a$ and $b$ is $\pi/6$,then find $|a \times b|^{2}$.

One side and one diagonal of a parallelogram are represented by $3 \hat{i}+\hat{j}-\hat{k}$ and $2 \hat{i}+\hat{j}-2 \hat{k}$ respectively. Then,the area of the parallelogram in square units is:

If the area of the parallelogram with $\bar{a}$ and $\bar{b}$ as two adjacent sides is $15$ square units,then the area (in square units) of the parallelogram,having $3 \bar{a} + 2 \bar{b}$ and $\bar{a} + 3 \bar{b}$ as two adjacent sides,is

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be the position vectors of the vertices $A, B, C$ respectively of $\triangle ABC$. The vector area of $\triangle ABC$ is:

If the equation of a straight line passing through a point with position vector $a$ and parallel to the vector $b$ is $r = a + t\,b$,where $t$ is a parameter,then what is the perpendicular distance from this line to a point with position vector $c$?

Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k} .$ Find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b},$ and $\vec{c} \cdot \vec{d}=15.$